English
For a self-adjoint a, the nonnegative spectrum bound implies a bound on the norm.
Русский
Для самопряженного a неверно, ограничение спектра на неотрицательное неравенство, ограничение нормы.
LaTeX
$$SpectrumRestricts (a) ContinuousMap.realToNNReal ↔ ‖algebraMap ℝ A t - a‖₊ ≤ t$$
Lean4
theorem smul_of_nonneg {A : Type*} [Ring A] [Algebra ℝ A] {a : A} (ha : SpectrumRestricts a ContinuousMap.realToNNReal)
{r : ℝ} (hr : 0 ≤ r) : SpectrumRestricts (r • a) ContinuousMap.realToNNReal :=
by
rw [SpectrumRestricts.nnreal_iff] at ha ⊢
nontriviality A
intro x hx
by_cases hr' : r = 0
· simp [hr'] at hx ⊢
exact hx.symm.le
· lift r to ℝˣ using IsUnit.mk0 r hr'
rw [← Units.smul_def, spectrum.unit_smul_eq_smul, Set.mem_smul_set_iff_inv_smul_mem] at hx
refine le_of_smul_le_smul_left ?_ (inv_pos.mpr <| lt_of_le_of_ne hr <| ne_comm.mpr hr')
simpa [Units.smul_def] using ha _ hx
